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```22 On the Association of Attributes in Statistics, &c.

" On the Association of Attributes in Statistics, with Examples
from the Material of the Childhood Society, &c." By G.
IJdny Yule, formerly Assistant Professor of Applied Mathe-
matics, University College, London. Commnnicated by Kael
Pearson-, F.K.S. Eeceived October 20,— Eead December 7,

1899.

(Abstract.)

The paper deals with the theory of association of attributes, i.e,^
invariable attributes, as opposed to the " correlation " of yariables.
Two attributes A and B are indeiaendent or unassociated if

(AB) = (A)(B)/N,

(A) being the frequency of the attribute A, (B) that of B, and (AB) the
frequency of the pair AB ; N being the total number of observations.
If this relation do not hold, they are '^ associated.'^

Section (I) of the paper is introductory, describing the subject-matter
and notation, which is essentially that of Jevons.* Calling a group

defined by n attributes ABGD N an 7ith-order group, Section (II)

deals with the fundamental problem of the number of independent nth
order frequencies that can be formed from m attributes ; i.e., the number
of such frequencies that must be given before the remaining frequencies
of the same order can be calculated. Certain extremely curious
relations are shown to hold in the special case of " equality of con-
traries," where all pairs of contrary frequencies (A) (a), (AB) (ocfS),
(ABC) (tty^y) are equal, cc being the contrary of A^ — i.e, not A~— and so
on.

Section (III) proceeds to the theory of association proper. The
function

^ _ (AB) (^13) - (A/?) jaB)

"^ '^ (AB) {af3) + {Ap) (ccB)

is proposed as a " coefficient of association." It is zero when the
attributes are independent, + 1 when all A^s are B or all B's are A,
and - 1 when all A's are jS or all /?'s are A, and thus measures the
approach towards "perfect association" in the same sort of way as
the correlation coefficient measures the approach towards perfect
correlation. The connection between correlation and association is
touched upon, and it is pointed out that one may form " partial "
coefficients of association (by limiting the extent of the universe of

* ^' On a Greneral System of Numerically Definite Beasoning," ' Manchester Lit.
and PMl. Soc.,' 1870, and " Pure Ijogio and other minor works" p, 173.

Data for the Prohlem of Evolution in Man. 23

discourse) roughly corresponding in their uses to partial coefficients of
correlation.

In Section (IV), the values of the Probable Errors, and the correla-
tions of the errors in the chief constants, are obtained. The probable
error of Q is

0-6745 —TT-a/ 7T^x+7Tm +-7-^ +

(AB)^(A^) ^ {aBy{af^Y

In Section (V), a series of miscellaneous illustrations are given
(association of smallpox attack rate and non-vaccination; association
between temper of husband and wife, inheritance of artistic faculty
&c., from Mr. Francis Galton's ' Natural Inheritance ' ; association
between vigour of offspring and crossing of parentage in plants from
Darwin's 'Cross and Self-fertilisation ').

In Section (VI), the " Association of defects in children and adults,"
is treated more at length as an example of the methods advocated, the
material being drawn mainly from the Eeport of the Committee on
the Scientific Study of Childhood. It is shown that the association
coefficient is almost uniformly higher for women than men, and for
children than adults. This last effect is however a mixed one, due
partly to selection, partly to change in the individual, and the
material available does not enable us to separate the partial effects.
These two laws of association appear to correspond to similar ones
for correlation ; women being more highly correlated than men, and
children than adults.

" Data for the Problem of Evolution in Man. III. — On the Mag-
nitude of certain Coefficients of Correlation in Man, &c." By
Kael Pearson, F.K.S., University College, London. Eeceived
November 20,- — Eead December 7, 1899.

1. This paper contains a number of data bearing on the correlation
of characters, &e., in man which have been worked out by my
collaborators during the last few years, and several of which seem
of considerable importance for problems relating to the evolution of
man. In each case the data were procured or reduced with a view to
answering some problem which had directly arisen during our inquiries
as to the action of natural selection on man. Questions as to the
alteration of correlation with growth or the influence of homogamy
on fertility demand definite answers before the general theory of the
influence of natural selection on a growing and reproductive popula-
tion can be effectively developed.

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